[SOLVED] CENG223 Take Home Exam 3

Use Fermat’s Little Theorem theorem the find (2²² + 4⁴⁴ + 6⁶⁶ + 8⁸⁰ + 10¹¹⁰) mod 11 ≡ ?

Note: Fermat’s Little Theorem is provided in our book (Kenneth H. Rosen, Discrete Mathematics and Its Applications), and it is a prerequisite for this question. This means that your solution have to use Fermat’s Little Theorem.

Question 2

Find gcd(5n + 3,7n + 4) and while doing that, show the steps of Euclid’s algorithm clearly.

Question 3

Let x be a prime number,

If m² = n² + kx where m, n, and k are integer numbers.

Show that x|(m + n) or x|(m − n).

Question 4

Show that for all n such that n ≥ 1 the following is true:

(1)

Note: You have to use mathematical induction to prove that.

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